Betti number signatures of homogeneous Poisson point processes

TL;DR

This paper investigates the expected Betti numbers of Poisson point processes, providing analytic formulas and simulations for low-intensity, small-radius regimes across different dimensions, linking topological measures with stochastic geometry.

Contribution

It introduces a tractable model for Betti numbers of Poisson processes and derives analytic expressions for their behavior in low-intensity, small-radius limits in 1D, 2D, and 3D.

Findings

Derived analytic expressions for Betti numbers in low-intensity, small-radius limits.

Provided simulation results validating the theoretical formulas.

Connected Betti number analysis with alpha-shapes from computational geometry.

Abstract

The Betti numbers are fundamental topological quantities that describe the k-dimensional connectivity of an object: B_0 is the number of connected components and B_k effectively counts the number of k-dimensional holes. Although they are appealing natural descriptors of shape, the higher-order Betti numbers are more difficult to compute than other measures and so have not previously been studied per se in the context of stochastic geometry or statistical physics. As a mathematically tractable model, we consider the expected Betti numbers per unit volume of Poisson-centred spheres with radius alpha. We present results from simulations and derive analytic expressions for the low intensity, small radius limits of Betti numbers in one, two, and three dimensions. The algorithms and analysis depend on alpha-shapes, a construction from computational geometry that deserves to be more widely known in the physics community.